All of the 5th grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$2.00$ each for students, and the group paid $$38.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$16.50$ each for teachers and $$10.50$ each for students, and the group paid $$150.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+2y = 38}$ ${16.5x+10.5y = 150}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-6y = -114}$ ${16.5x+10.5y = 150}$ Add the top and bottom equations together. $ 4.5y = 36 $ $ y = \dfrac{36}{4.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5.5x+2y = 38}$ to find $x$ ${5.5x + 2}{(8)}{= 38}$ $5.5x+16 = 38$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {16.5x+10.5y = 150}$ and get the same answer for $x$ ${16.5x + 10.5}{(8)}{= 150}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.